Solving Linear Systems Statistics Study Guide - … · Unit 1 - Solving Linear Systems ... Answer questions 1 - 5. 1.5 Exercises, ... putting these word problems into equations.

Adult Basic EducationMathematics

Mathematics 2104A

Solving Linear SystemsStatistics

Study GuidePrerequisites: Mathematics 1104A, 1104B, 1104C

Credit Value: 1

Text: Mathematics 10. Alexander and Kelly; Addison-Wesley, 1998.Mathematics 11. Alexander and Kelly; Addison-Wesley, 1998.

Required Mathematics Courses [Degree and Technical Profile/Business-Related College Profile]

Mathematics 1104AMathematics 1104BMathematics 1104CMathematics 2104AMathematics 2104BMathematics 2104CMathematics 3104AMathematics 3104BMathematics 3104C

Table of Contents

To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vIntroduction to Mathematics 2104A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vResources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vStudy Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viRecommended Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Unit 1 - Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 1

Unit 2 - Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 8

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 16

To the Student

Study Guide Mathematics 2104Av

I. Introduction to Mathematics 2104A

In this course, you will solve linear systems of equations using addition and subtraction. Usingthis technique to solve a system of equations with three or more unknowns may be verycumbersome. Instead, you will learn to use matrices for these complex situations. You willwrite systems of equations in matrix form and, using technology, find the inverse matrix andthen solve the system. The material for this topic is found in Mathematics 11 and Appendix A,at the end of the Study Guide. You will look at linear systems and classify them as consistent(one solution or many solutions) or inconsistent (no solution).

You will also be introduced to inferential statistics and examine various sampling methods andpotential sources of bias associated with these methods. You will make inferences aboutsamples based on population data and make inferences about populations based on data fromsamples.

II. Resources

You will require the following:

• Addison Wesley Mathematics 10, Western Canadian edition Textbook• Addison Wesley Mathematics 11, Western Canadian edition Textbook• Scientific calculator • graph paper• Access to a TI-83 Plus graphing calculator (see your instructor)

and/or Graphmatica or Winplot graphing software

Notes concerning the textbook:

Glossary: Knowledge of mathematical terms is essential to understand concepts and correctlyinterpret questions. Written explanations will be part of the work you submit for evaluation, andappropriate use of vocabulary will be required.

Your text for this course includes a Glossary where definitions for mathematical terms are found.Be sure you understand such definitions and can explain them in your own words. Whereappropriate, you should include examples or sketches to support your definitions.

Examples: You are instructed to study carefully the Examples in each section and see yourinstructor if you have any questions. These Examples provide full solutions to problems thatcan be of great use when answering assigned Exercises.

To the Student

Study Guide Mathematics 2104Avi

Notes concerning technology:

It is important that you have a scientific calculator for your individual use. Ensure that thecalculator used has the word “scientific” on it as there are calculators designed for calculation inother areas such as business or statistics which would not have the functions needed for study inthis area. Scientific calculators are sold everywhere and are fairly inexpensive. You shouldhave access to the manual for any calculator that you use. It is a tool that can greatly assist thestudy of mathematics but, as with any tool, the more efficient its use, the better the progress.

You will require access to some sort of technology in order to meet some of the outcomes in thiscourse. Since technology has become a significant tool in the study of Mathematics, yourtextbook encourages you to become proficient in its use by providing you with step-by-stepexercises that will teach you about the useful functions of the TI-83 Plus Graphing calculator.See your instructor concerning this. Please note that a graphing calculator is not essential forsuccess in this course but it is useful.

While graphing calculators and graphing software (Graphmatica or Winplot) are useful tools,they cannot provide the same understanding that comes from working paper and pencilexercises.

III. Study Guide

This Study Guide is required at all times. It will guide you through the course and you shouldtake care to complete each unit of study in the order given in this Guide. Often, at the beginningof each unit, you will be instructed to see your instructor for Prerequisites exercises. Please donot skip this step! It should take only a few minutes for you and your instructor to discover what,if any, prerequisite skills need review.

To be successful, you should read the References and Notes first and then, when indicated bythe || symbols, complete the Work to Submit problems. Many times you will be directed tosee your instructor, and this is vital, especially in a Mathematics course. If you have only a hazyidea about what you just completed, nothing will be gained by continuing on to the next set ofproblems.

To the Student

Study Guide Mathematics 2104Avii

Reading for this Unit: In this box, you will find the name of the text, and the chapters, sections and pages usedto cover the material for this unit. As a preliminary step, skim the referenced section, looking at the name of thesection, and noting each category. Once you have completed this overview, you are ready to begin.

References and Notes Work to Submit

This left hand column guides you through thematerial to read from the text.

It will also refer to specific Examples foundin each section. You are directed to studythese Examples carefully and see yourinstructor if you have any questions. TheExamples are important in that they not onlyexplain and demonstrate a concept, but alsoprovide techniques or strategies that can beused in the assigned questions.

The symbols || direct you to the column onthe right which contains the work to completeand submit to your instructor. You will beevaluated on this material.

Since the answers to Discussing the Ideasand Communicating the Ideas are not foundin the back of the student text, you must havethese sections corrected by your instructorbefore going on to the next question.

This column will also contain general Noteswhich are intended to give extra information and are not usually specific to any onequestion.

There are four basic categories included in this column thatcorrespond to the same categories in the sections of the text.They are Investigate, Discussing the Ideas, Exercises, andCommunicating the Ideas.

Investigate: This section looks at the thinking behind newconcepts. The answers to its questions are found in the back ofthe text.

Discussing the Ideas: This section requires you to write aresponse which clarifies and demonstrates your understandingof the concepts introduced. The answers to these questions arenot in the student text and will be provided when you see yourinstructor.

Exercises: This section helps to reinforce your understanding ofthe concepts introduced. There are three levels of Exercises:A: direct application of concepts introduced B: multi-step problem solving and some real-life situationsC: problems of a more challenging natureThe answers to the Exercises questions are found in the back ofthe text.

Communicating the Ideas: This section helps confirm yourunderstanding of the lesson of the section. If you can write aresponse, and explain it clearly to someone else, this means thatyou have understood the topic. The answers to these questionsare not in the student text and will be provided when you seeyour instructor

This column will also contain Notes which give informationabout specific questions.

To the Student

Study Guide Mathematics 2104Aviii

IV. Recommended Evaluation

Written Notes 10%Assignments 10%Test(s) 30%Final Exam (entire course) 50%

100%The overall pass mark for the course is 50%.

Unit 1 - Solving Linear Systems

Study Guide Mathematics 2104APage 1

To meet the objectives of this unit, students should complete the following:

Reading for this unit: Mathematics 11 Chapter 5: Section 5.2: pages 310 - 318

Section 5.3: pages 319 - 322Section 5.6: pages 337 - 343

Mathematics 2104A Study Guide: Appendix: pages 17 - 26

References and Notes

Read Section 5.2.

Answer the following questions. ||

Work to Submit

1.1 Investigate, page 310Answer questions 1 - 6.

1.2 Discussing the Ideas, page 315Answer questions 1 - 4.

1.3 Exercises, pages 315 - 318Answer questions 1 - 3.(See notes below on these questions.)

Answer questions 5 - 8.(See note below on questions 7 and 8.)

Answer questions 12 and 19.(See note below on question 12.)

Question 1: You don’t have to solve each system to see if(!1, 1) is a solution.

Questions 2 and 3: Don’t forget to multiply both sides ofthe equation by a constant.

Questions 7 and 8: These exercises are very similar toExample 4.

Question 12: Expand the equations in g) and h) andrearrange the terms in all equations so that it will be easierto add or subtract vertically.




Read Section 5.3.

Study and work throughExamples 1, 2 and 3.


Read Section 5.6.

Visualizing, page 337, gives anexplanation on what a linearequation in three variablesrepresents.

Work to Submit


1.5 Exercises, pages 321 and 322Answer questions 1 - 3.(See note below on question 1.)

Answer questions 4 and 5.(See note below on these questions.)

Answer questions 6 and 7.

Question 1: There is more than one way to answer thisquestion. You could express each equation in the form y = mx + b and look at the m and b values.

Example 3 gives another technique that you could follow.

Questions 4 and 5: Make sure that you understand themeaning of consistent and inconsistent before youcomplete these exercises.

1.6 See your instructor for Prerequisites exercises.




Answer the following question: ||

Study Examples 1, 2 and 3.Work though all of the givensteps.

The initial equation in Example3 is a quadratic. However, youwill note that when values aresubstituted for n, you will have alinear equation with threeunknowns.

Answer the following questions.||

Work to Submit

1.7 Investigate, page 337Answer questions 1, 2 and 3.


1.9 Exercises, page 341 - 343Answer questions 1 - 8.(See note below on question 8.)

Answer questions 9 -11.(See note below on these questions.)

Answer question 12.(See note below on question 12.)

Question 8: Try this: Write an expression ax + by +czwith any non-zero values of a, b, and c that you choose. Substitute the given values of x, y and z in the equationand simplify. Write an equation with the expression on theleft side and the simplification on the right side.




You will use matrices to solvesystems of linear equations. Youwill use the Appendix and otherresources provided by yourinstructor to complete thissection.

You may need to review thework that you did on matrices inMathematics 1104C before youstart.

You must have access to a TI-83for all of this topic.

Read Appendix, page 17.


Work to Submit

Questions 9 -11: See your instructor if you have difficultyputting these word problems into equations.

Question 12: You can use a graphing calculator to graphthe function that you find. Use the CALCULATE menu to confirm your answer inpart b. Use the window settings G2 # X # 10 and G1000 # Y # 40 000.

1.10 Exercises, page 17Answer questions 1 - 3.(See note below on question 3.)

Question 3: You should find that AI = IA. However,recall that, in general, multiplication of matrices is notcommutative. In other words, given two matrices A andB, AB … BA.




Read Appendix, page 18.

You will practise finding theinverse of square matrices. Thisis a critical step when usingmatrices to solve systems ofequations.


Read Appendix, pages 20 and21. Carefully study Examples 2and 3. Make sure that youunderstand how to write asystem of equations in matrixform.

On the bottom of pages 20 and21, you are asked to find (A)(X). When you do this multiplication,the result should be the left handside of the original system ofequations.

Note that you must have thesame number of equations asthere are variables before youcan solve a system of equations.

Work to Submit

1.11 Exercises, page 19Answer questions 4 and 5.(See note below on question 5.)

Question 5: In order to determine whether or not B = AG1,find AB or BA. If the result is the identity matrix, thenyou know that they are inverses of each other. (Or, B = AG1 and A = BG1 )




Read Appendix, pages 22 to 25.

Carefully study Examples 4 and5.

Ask your instructor for moreexplanation if you do notunderstand the importance of theinverse matrix and its purposewhen solving a system ofequations.

Also, see your instructor if youare having difficulty using theTI-83 to work through Examples4 and 5.


Study Example 6, page 26.


Work to Submit

1.12 Exercises, page 25Answer questions 6 and 7.

1.13 Exercises, page 26Answer questions 8, 9 and 10.

Unit 2 - Statistics


To meet the objectives of this unit, students should complete the following:

Reading for this unit: Mathematics 10Chapter 9: Section 9.1: pages 530 - 535

Mathematics File: pages 536 and 537Section 9.2: pages 538 - 542Mathematics File: page 543Linking Ideas: page 544Section 9.3: pages 545 - 554Section 9.4: pages 555 - 559


Read Section 9.1.


Read page 531, study Exampleon page 532 and answer thefollowing questions. ||

See your instructor to haveDiscussing the Ideas correctedbefore moving on to theExercises.

Work to Submit

2.1 Investigate, page 530Answer questions 1 - 5.

2.2 Define the following terms:i) populationii) sampleiii) bias


Unit 2 - Statistics



Read Mathematics File, ‘Usinga Graphing Calculator’ only.

Your instructor will have theprogram called RANDNUM. You can enter this program onyour calculator. Alternatively,you can use the unit-to-unit linkcable which comes with the TI-83 calculator to copy thisprogram from one calculator toanother.

Mathematics File, pages 536and 537, lists four different waysto generate random numbers. You will be using the graphingcalculator only, although youshould read the other 3 methods.

Work to Submit

2.4 Exercises, pages 533 and 534Answer questions 1 - 11.(See note below on question 6.)

Question 6: Include segments of the population thatwould bias a survey in any direction.

2.5 Communicating the Ideas, page 535

2.6 Define the term random numbers.

Unit 2 - Statistics



The TI-83 simulates the resultsof experiments with objects suchas spinners, dice or coins.

Press MATH, PRB, randInt, toget: randInt (lowest number,highest number, number oftrials).

This command will generate alist of random integers that aregreater than or equal to the lowerboundary and less than or equalto the upper boundary. Thelength of the list corresponds tothe number of trials requested.

For example, randInt (1, 10, 5)will create a list of 5 randomdigits $1, but #10. To simulatethe rolling of a die, use randIntwith a lower boundary of 1 andan upper boundary of 6.

To simulate the flipping of acoin, set the lower boundary to 0and the upper to 1, with 0representing heads and 1representing tails.


Work to Submit

2.7 Mathematics Files, pages 536 and 537Answer questions 1, 2, 3, 4 and 5.

Unit 2 - Statistics



Read Section 9.2.

Study Example on page 539.


See your instructor aboutDiscussing the Ideas beforecompleting the Exercises.

Work to Submit

2.8 See your instructor for Prerequisites problemsbefore completing this section.

2.9 Describe the following sampling methods:i)) Cluster sampleii) Convenience sampleiii) Self-selected sampleiv) Simple random samplev) Stratified random samplevi) Systematic sample

2.10 Discussing the Ideas, page 539Answer questions 1, 2, 3 and 4.

2.11 Exercises, pages 540 - 542Answer questions 1- 6, 9, 10, 12, 13 and 14.(See note below on questions 13 and 14.)

Questions 13 and 14: Choose one issue from either ofthose questions and design a survey.

Unit 2 - Statistics



Before completing Section 9.3,read Mathematics File on page543. Do not make the samplingbox, but read it so that you willhave an understanding of its use.

Read Linking Ideas, question 1on page 544. Make sure that youhave the SMPLSIM3 programworking on your calculatorbefore you start Section 9.3.

Answer the following question.||

Read Section 9.3.

In this section you will knowinformation about a givenpopulation. You will simulatethe sampling of this populationand put the results in boxplots.


Work to Submit

2.12 Linking Ideas, page 544Answer question 1.

2.13 Define the following terms:i) binomial outcomeii) binomial population

2.14 Investigate, page 545Answer questions 1, 3, 5 and 6.

Unit 2 - Statistics



Do not confuse the percents usedin the sentence following Step 4on page 547. The 90% refers tothe probability of somethinghappening (in this case, therebeing between 12 and 17 markeditems); while the 70% refers tothe percent of marked items inthe population.


Study Examples 1 and 2. Workthrough the given solutions.

If you place a ruler vertically onthe boxplots it will be easier toread the values.


Work to Submit

2.15 Investigate, page 547Answer questions 1, 2, 3 and 4.(See note below on question 1.)

Question 1: Generate 100 samples with 10% of beadsmarked and 100 samples with 30% of beads marked.

2.16 Discussing the Ideas, page 549Answer questions 1, 2 and 3.

Unit 2 - Statistics



Read Section 9.4.

In this section the procedure ofSection 9.3 is reversed. You aregiven the number of markeditems in a sample, the samplesize and then you will interpretboxplots to make an inferenceabout the population.

On page 556, study Exampleand work through the givensolution.

You will notice that the intervalfor the estimated percent in thepopulation narrows as the samplesize increases.

Work to Submit

2.17 Exercises, page 549 - 551Answer questions 1 - 7.(See note below on these questions.)

Answer questions 9, 10, 11, 14 and 15.

Note: You must use the appropriate boxplots from pages552 - 554 to complete these exercises.

You should calculate the percent of the sample to checkwhether your estimate is reasonable.

Unit 2 - Statistics



You can check your inference in

Example a) by calculating as1240

a percent, 30%. Why is thispercent close to the median(middle number) of the percentsestimated for the boxplots?


Work to Submit

2.18 Exercises, page 557Answer questions 1 - 9 and 12.

Appendix A


MatricesIdentity Matrix

The identity element for real number multiplication is 1. In matrix theory there is an IdentityMatrix which, when multiplied by any other matrix, leaves that matrix unchanged.

Exercises:

1. Using Guess and Check, attempt to generate the 2 × 2 Identity Matrix, I, that when multiplied

by leaves this matrix unchanged.2 31 4−⎡

⎣⎢

⎤

⎦⎥

2. Use the TI-83 to determine the 3 × 3 Identity Matrix.

3. Create any 2 × 2 matrix and name it A. Explore the solutions to AI and IA. Write a fewsentences explaining the answers you obtained.

You should notice that the identity matrix must be square.

If A = , the identity matrix, I, is .

4 3 10 1 26 9 1−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The identity matrix is a square matrix with 1's on the main diagonal. (top left to bottom right)and all other elements are 0.


Inverse Matrix

Any real number (except 0) has a multiplicative inverse. A number multiplied by its inverse will

yield 1, the multiplicative identity. For example, the multiplicative inverse of is . We37

73

know that . Similarly, most square matrices have inverses. The inverse of matrix A37

73

1× =

is denoted by A-1. A matrix multiplied by its inverse will yield the identity matrix.

AA!1 = I and A!1A = I

The inverse of a matrix can be calculated by hand and, except for a 2 × 2 matrix, is very timeconsuming. For this course, we will find the inverse of matrices by using a TI-83 graphingcalculator.

Use a TI-83 to get the inverse of 2 × 2 and 3 × 3 or any square matrix.

Example 1:

A = . When you enter A in your TI-83, you should see the following display:4 31 1⎡

⎣⎢

⎤

⎦⎥

To find A!1, press: [2nd], [MATRIX], [ENTER],[ x!1], [ENTER]. You should see the following display:

Multiply AA!1 or A!1A. You will get the identity matrix.


Exercises:

4. Use a TI-83 to get the inverse of each of the following. In each case, show that AA!1 = I and A!1A = I.

a) A = b) A = 4 31 1⎡

⎣⎢

⎤

⎦⎥

1 23 7⎡

⎣⎢

⎤

⎦⎥

c) A = d) A = −−⎡

⎣⎢

⎤

⎦⎥

3 22 1

1 32 5⎡

⎣⎢

⎤

⎦⎥

e) A = f) A =

1 3 52 0 11 2 4

−

−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 2 04 3 12 1 3

5. In exercises a) - d), determine whether or not B = A!1.

a) A = B = 5 22 1

−−⎡

⎣⎢

⎤

⎦⎥

1 22 5⎡

⎣⎢

⎤

⎦⎥

b) A = B = 3 45 7

−−

⎡

⎣⎢

⎤

⎦⎥

7 45 2

−−

⎡

⎣⎢

⎤

⎦⎥

c) A = B =

1 2 32 5 71 3 5

−−

− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

4 1 13 2 11 1 1

−− −− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

d) A = B =

1 1 33 4 82 3 4

−−

− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

8 5 44 2 11 1 1

− −− −

−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


Using matrices to solve a system of equations

Before we can use matrices to solve a system of equations, we must be able to write the systemin matrix form. To do this, we write one matrix containing the coefficients of the variables, onematrix containing the variables, and one matrix containing the constant terms.

Example 2:

Given the following system of equations:

3r + s + t = 1.1!2r + 2s !3t = 3.4r + 5s +2t = 5.3

The coefficient matrix, A, is .

3 1 12 2 3

1 5 2− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The variable matrix, X, is

rst

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The values matrix, B, is

113453

.

.

.

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The matrix form could be written (A) (X) = B, which gives us ;

3 1 12 2 3

1 5 2

113453

− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

rst

.

.

.

Multiply: (A)(X). What is the result?


To write the matrix form of a system of equations, there must be the same number of rows asthere are variables. If any variables or equations are missing, they must be filled in with zerocoefficients.

Example 3:

The system, 2x + 3y = 5 !2y + z = !1

x + z = 2

when written with all variables included in every equation becomes:

2 3 0 50 2 1

0 2

x y zx y z

x y z

+ + =− + = −

+ + =

For this example, A = , X = and B =

2 3 00 2 11 0 1

−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

51

2−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

The matrix form could be written (A) (X) = B, which is:

2 3 00 2 11 0 1

51

2−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

Again, what is the result when you multiply (A)(X)?


Look at the matrix form AX = B.

If you multiply both sides by the inverse, A!1, you will get

(A!1) (A)(X) = (A!1)(B)

( I ) (X) = (A!1)(B)

X = (A!1)(B)

This will give a one column matrix, X, whose elements are the solution to the system.

Solving a Matrix Equation

Before we begin solving matrix equations, think back to solving regular equations such as:

34

6x =

In that case, we multiplied both sides of the equation by the inverse of the coefficient whichgave:

( )43

34

43

6x⎛⎝⎜

⎞⎠⎟ =

1x = 8 x = 8

Notice that we wanted to get the coefficient of x to be 1, the identity for multiplication.

Now let us apply this idea to matrix equations. Even though the matrix form of a system isintroduced to help solve more complex systems, we will start by applying the technique to asimple system in two variables. The technique would be the same regardless of the number ofvariables.


Example 4:Let us consider the following system:

!2x = y !14x + y = !5

Rewrite the system as: !2x ! y = !1 4x + y = !5

Write the system in matrix form:

− −⎡

⎣⎢

⎤

⎦⎥⎡

⎣⎢

⎤

⎦⎥ =

−−⎡

⎣⎢

⎤

⎦⎥

2 14 1

15

xy

We now multiply both sides by the inverse of the coefficient matrix, just as we multiplied thecoefficient and inverse of the coefficient in the equation above.

The inverse of a matrix is written with a superscript of !1, thus the inverse of our coefficientmatrix is written as:

, and the equation can be written as:− −⎡

⎣⎢⎤

⎦⎥

−2 14 1

1

− −⎡

⎣⎢

⎤

⎦⎥

− −⎡

⎣⎢

⎤

⎦⎥⎡

⎣⎢⎤

⎦⎥ =

− −⎡

⎣⎢

⎤

⎦⎥

−−⎡

⎣⎢

⎤

⎦⎥

− −2 14 1

2 14 1

2 14 1

15

1 1xy

Since the two matrices on the left are inverses of each other, their product will give the identitymatrix and our equation will look like this:

xy⎡

⎣⎢⎤

⎦⎥ =

− −⎡

⎣⎢

⎤

⎦⎥

−−⎡

⎣⎢

⎤

⎦⎥

−2 14 1

15

1


To solve the equation, you need to determine the following inverse: − −⎡

⎣⎢

⎤

⎦⎥

−2 14 1

1

Use your TI-83 calculator, which can calculate the inverse of a matrix.

The inverse matrix will be:. .5 5

2 1− −⎡

⎣⎢

⎤

⎦⎥

You now have the inverse matrix that you desired. Write:

− −⎡

⎣⎢

⎤

⎦⎥ =

− −⎡

⎣⎢

⎤

⎦⎥

−2 14 1

5 52 1

1 . .

Solving the original equation gives:

xy⎡

⎣⎢⎤

⎦⎥ =

− −⎡

⎣⎢

⎤

⎦⎥

−−⎡

⎣⎢

⎤

⎦⎥

−2 14 1

15

1

xy⎡

⎣⎢⎤

⎦⎥ = − −

⎡

⎣⎢

⎤

⎦⎥−−⎡

⎣⎢

⎤

⎦⎥

. .5 52 1

15

xy⎡

⎣⎢⎤

⎦⎥ =

× − + × −− × − + − × −

⎡

⎣⎢

⎤

⎦⎥

(. ) (. )( ) ( )

5 1 5 52 1 1 5

xy⎡

⎣⎢⎤

⎦⎥ =

−⎡

⎣⎢

⎤

⎦⎥

37

So the solution to the original system is x = !3 and y = 7. You should check this by substitutinginto the original system.


Example 5:

Given: x & 2y = 73x + 4y = 1,

solve for x and y. Write the system in matrix form first.

A = , B = , X = 1 23 4

−⎡

⎣⎢

⎤

⎦⎥

71⎡

⎣⎢⎤

⎦⎥

xy⎡

⎣⎢⎤

⎦⎥

Write: (A)(X) = B

X = (A!1)(B)

Enter Matrix A.

Enter Matrix B.

Press [2nd] [QUIT] and then the following:

Press [Matrix A], [ENTER],[x!1] [×] [MATRIX] [?][ 2:B],[ENTER]

You should see the following window:

The solution is (3, !2).

Exercises:

Solve the following systems using matrices:

6. 3x + 2y = 2 7. 3x + 2y = 194x + 5y = 12 5x ! 7y = 5

Matrices become more practical as the number of variables increases.


Example 6:

To solve the following system, you will get the windows as shown:

4x + y + z = 52x !y + 2z =10x - 2y - z = 2

Work through the necessary steps on your TI-83 to solve thissystem.

Exercises:

Solve the following systems using matrices:

8. x & 3y & 2z = 93x + 2y + 6z = 204x &y + 3z = 25

9. x + y + z = !22x & y ! z = !13x + 2y + 4z = !15

10. 3x + 2y ! 5z = !9x ! 3y + 4z = 232x + y !3z = !4

Solving Linear Systems Statistics Study Guide - … · Unit 1 - Solving Linear Systems ... Answer questions 1 - 5. 1.5 Exercises, ... putting these word problems into equations.

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